Basic Math Examples

$\frac{\sqrt[3]{9}}{\sqrt[5]{27}}$

Step 1

Multiply $\frac{\sqrt[3]{9}}{\sqrt[5]{27}}$ by $\frac{{\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{4}}$ .

$\frac{\sqrt[3]{9}}{\sqrt[5]{27}} \cdot \frac{{\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{4}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt[3]{9}}{\sqrt[5]{27}}$ by $\frac{{\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{4}}$ .

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{\sqrt[5]{27} {\sqrt[5]{27}}^{4}}$

Step 2.2

Raise $\sqrt[5]{27}$ to the power of $1$ .

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{1} {\sqrt[5]{27}}^{4}}$

Step 2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{1 + 4}}$

Step 2.4

Add $1$ and $4$ .

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{{\sqrt[5]{27}}^{5}}$

Step 2.5

Rewrite ${\sqrt[5]{27}}^{5}$ as $27$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{27}$ as $27^{\frac{1}{5}}$ .

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{{(27^{\frac{1}{5}})}^{5}}$

Step 2.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{27^{\frac{1}{5} \cdot 5}}$

Step 2.5.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{27^{\frac{5}{5}}}$

Step 2.5.4

Cancel the common factor of $5$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{27^{\frac{5}{5}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{27^{1}}$

Step 2.5.5

Evaluate the exponent.

$\frac{\sqrt[3]{9} {\sqrt[5]{27}}^{4}}{27}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[5]{27}}^{4}$ as $\sqrt[5]{27^{4}}$ .

$\frac{\sqrt[3]{9} \sqrt[5]{27^{4}}}{27}$

Step 3.2

Raise $27$ to the power of $4$ .

$\frac{\sqrt[3]{9} \sqrt[5]{531441}}{27}$

Step 3.3

Rewrite $531441$ as $9^{5} \cdot 9$ .

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Step 3.3.1

Factor $59049$ out of $531441$ .

$\frac{\sqrt[3]{9} \sqrt[5]{59049 (9)}}{27}$

Step 3.3.2

Rewrite $59049$ as $9^{5}$ .

$\frac{\sqrt[3]{9} \sqrt[5]{9^{5} \cdot 9}}{27}$

Step 3.4

Pull terms out from under the radical.

$\frac{\sqrt[3]{9} \cdot 9 \sqrt[5]{9}}{27}$

Step 3.5

Combine exponents.

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Step 3.5.1

Rewrite the expression using the least common index of $15$ .

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Step 3.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9}$ as $9^{\frac{1}{3}}$ .

$\frac{9 (9^{\frac{1}{3}} \sqrt[5]{9})}{27}$

Step 3.5.1.2

Rewrite $9^{\frac{1}{3}}$ as $9^{\frac{5}{15}}$ .

$\frac{9 (9^{\frac{5}{15}} \sqrt[5]{9})}{27}$

Step 3.5.1.3

Rewrite $9^{\frac{5}{15}}$ as $\sqrt[15]{9^{5}}$ .

$\frac{9 (\sqrt[15]{9^{5}} \sqrt[5]{9})}{27}$

Step 3.5.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{9}$ as $9^{\frac{1}{5}}$ .

$\frac{9 (\sqrt[15]{9^{5}} \cdot 9^{\frac{1}{5}})}{27}$

Step 3.5.1.5

Rewrite $9^{\frac{1}{5}}$ as $9^{\frac{3}{15}}$ .

$\frac{9 (\sqrt[15]{9^{5}} \cdot 9^{\frac{3}{15}})}{27}$

Step 3.5.1.6

Rewrite $9^{\frac{3}{15}}$ as $\sqrt[15]{9^{3}}$ .

$\frac{9 (\sqrt[15]{9^{5}} \sqrt[15]{9^{3}})}{27}$

Step 3.5.2

Combine using the product rule for radicals.

$\frac{9 \sqrt[15]{9^{5} \cdot 9^{3}}}{27}$

Step 3.5.3

Multiply $9^{5}$ by $9^{3}$ by adding the exponents.

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Step 3.5.3.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 \sqrt[15]{9^{5 + 3}}}{27}$

Step 3.5.3.2

Add $5$ and $3$ .

$\frac{9 \sqrt[15]{9^{8}}}{27}$

Step 3.6

Raise $9$ to the power of $8$ .

$\frac{9 \sqrt[15]{43046721}}{27}$

Step 3.7

Rewrite $43046721$ as $3^{15} \cdot 3$ .

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Step 3.7.1

Factor $14348907$ out of $43046721$ .

$\frac{9 \sqrt[15]{14348907 (3)}}{27}$

Step 3.7.2

Rewrite $14348907$ as $3^{15}$ .

$\frac{9 \sqrt[15]{3^{15} \cdot 3}}{27}$

Step 3.8

Pull terms out from under the radical.

$\frac{9 (3 \sqrt[15]{3})}{27}$

Step 4

Multiply $9$ by $3$ .

$\frac{27 \sqrt[15]{3}}{27}$

Step 5

Cancel the common factor of $27$ .

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Step 5.1

Cancel the common factor.

$\frac{27 \sqrt[15]{3}}{27}$

Step 5.2

Divide $\sqrt[15]{3}$ by $1$ .

$\sqrt[15]{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[15]{3}$

Decimal Form:

$1.07598962 \dots$